Theorem elsymdifxor

Description: Membership in a symmetric difference is an exclusive-or relationship. (Contributed by David A. Wheeler, 26-Apr-2020) (Proof shortened by BJ, 13-Aug-2022)

	Ref	Expression
Assertion	elsymdifxor	⊢ ( 𝐴 ∈ ( 𝐵 △ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elsymdif	⊢ ( 𝐴 ∈ ( 𝐵 △ 𝐶 ) ↔ ¬ ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶 ) )
2		df-xor	⊢ ( ( 𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶 ) ↔ ¬ ( 𝐴 ∈ 𝐵 ↔ 𝐴 ∈ 𝐶 ) )
3	1 2	bitr4i	⊢ ( 𝐴 ∈ ( 𝐵 △ 𝐶 ) ↔ ( 𝐴 ∈ 𝐵 ⊻ 𝐴 ∈ 𝐶 ) )